Orbits Of Maps Research Articles

KAM quasi-periodic tori for the dissipative spin–orbit problem

We provide evidence of the existence of KAM quasi-periodic attractors for a dissipative model in Celestial Mechanics. We compute the attractors extremely close to the breakdown threshold.We consider the spin–orbit problem describing the motion of a triaxial satellite around a central planet under the simplifying assumption that the center of mass of the satellite moves on a Keplerian orbit, the spin-axis is perpendicular to the orbit plane and coincides with the shortest physical axis. We also assume that the satellite is non-rigid; as a consequence, the problem is affected by a dissipative tidal torque that can be modeled as a time-dependent friction, which depends linearly upon the velocity.Our goal is to fix a frequency and compute the embedding of a smooth attractor with this frequency. This task requires to adjust a drift parameter.We have shown in Calleja et al. (2020) that it is numerically efficient to study Poincaré maps; the resulting spin–orbit map is conformally symplectic, namely it transforms the symplectic form into a multiple of itself. In Calleja et al. (2020), we have developed an extremely efficient (quadratically convergent, low storage requirements and low operation count per step) algorithm to construct quasi-periodic solutions and we have implemented it in extended precision. Furthermore, in Calleja et al. (2020) we have provided an “a-posteriori” KAM theorem that shows that if we have an embedding and a drift parameter that satisfy the invariance equation up to an error which is small enough with respect to some explicit condition numbers, then there is a true solution of the invariance equation. This a-posteriori result is based on a Nash–Moser hard implicit function theorem, since the Newton method incurs losses of derivatives.The goal of this paper is to provide numerical calculations of the condition numbers and verify that, when they are applied to the numerical solutions, they will lead to the existence of the torus for values of the parameters extremely close to the parameters of breakdown. Computing reliably close to the breakdown allows to discover several interesting phenomena, which we will report in Calleja et al. (2020).The numerical calculations of the condition numbers presented here are not completely rigorous, since we do not use interval arithmetic to estimate the round off error and we do not estimate rigorously the truncation error, but we implement the usual standards in numerical analysis (using extended precision, checking that the results are not affected by the level of precision, truncation, etc.). Hence, we do not claim a computer-assisted proof, but the verification is more convincing than a standard numerics. We hope that our work could stimulate a computer-assisted proof.

Synthesis, Characterization, Crystal Structure, Hirshfeld surface analysis and DFT studies of novel compounds based on the methoxynaphthalene ring

• Synthesis of novel series of methoxynaphthalene derivatives. • Confirmation of their chemical structures by spectroscopic techniques: IR, 1 H, 13 C NMR, LC/HRMS. • The structures of 5, 6, 8, 1 1, 12, 14 and 17 have been established by single crystal X-ray diffraction. • Intermolecular interactions in crystal packing using Hirshfeld surface analysis was also provided. • DFT calculation have been investigated and compared with X-ray results. Polysubstituted naphthalenes and their analogues have been found in nature or synthesized, and have attracted increasing attention due to their numerous pharmacological and biological activities. The present study describes a synthesis of novel series of methoxynaphthalene derivatives using the Stobbe condensation reaction method. The chemical structures of these compounds were confirmed by spectroscopic techniques such as 1 H and 13 C NMR, IR, LC-HRMS spectral data and X-ray crystallography. The supramolecular assembly in the structures reported here is dominated by C—H···O hydrogen bonds for compounds 5, 6, 8, 11, 14 and 17 and O—H···O for compound 12 . The molecules are also arranged by C–H···π interaction and/or π—π stacking, which are responsible for the formation and stability of the molecular assemblies. Hirshfeld surface analysis was also performed in order to obtain reliable structural information in concurrence with experimental results, and intermolecular interactions that exist inside the crystal have been investigated. Additionally, DFT calculations have been used to analyze the electronic and geometric frontier molecular orbital and Molecular Electrostatic Potential map analyses of the compounds were produced using the optimized structures. This manuscript report the synthesis, X-ray characterization and theoretical study of novel series of methoxynaphthalene derivatives.

Orbital Mapping of Semiconducting Perylenes on Cu(111).

Semiconducting O-doped polycyclic aromatic hydrocarbons constitute a class of molecules whose optoelectronic properties can be tailored by acting on the π-extension of the carbon-based frameworks and on the oxygen linkages. Although much is known about their photophysical and electrochemical properties in solution, their self-assembly interfacial behavior on solid substrates has remained unexplored so far. In this paper, we have focused our attention on the on-surface self-assembly of O-doped bi-perylene derivatives. Their ability to assemble in ordered networks on Cu(111) single-crystalline surfaces allowed a combination of structural, morphological, and spectroscopic studies. In particular, the exploitation of the orbital mapping methodology based on angle-resolved photoemission spectroscopy, with the support of scanning tunneling microscopy and low-energy electron diffraction, allowed the identification of both the electronic structure of the adsorbates and their geometric arrangement. Our multi-technique experimental investigation includes the structure determination from powder X-ray diffraction data for a specific compound and demonstrates that the electronic structure of such large molecular self-assembled networks can be studied using the reconstruction methods of molecular orbitals from photoemission data even in the presence of segregated chiral domains.

Interaction of krypton and xenon with sodium and activated charcoal: Identification and modeling using gas chromatography and density functional theory

For the online detection of failed fuel pin in Prototype Fast Breeder Reactor (PFBR), Failed Fuel Localization Module (FFLM) facility is designed to probe the sodium (Na) coolant. Fission Gas Sparging Module (FGSM) is retrofitted with FFLM to localize failed fuel subassembly during the dry rupture of the fuel pin. FGSM is used to purge gaseous fission products (krypton (Kr) and xenon (Xe)) from Na, using argon (Ar). In this work, the efficiency of separating inactive Kr and Xe noble gases with Ar purge out of liquid Na, by their collection onto activated charcoal at cryogenic temperatures and the consequent release at high temperatures into the gas samplers was studied using gas chromatography (GC). Studies of the recovery of Xe/Kr separately with and without Na matrix in the FGSM under both static and dynamic conditions were undertaken by sequentially collecting these probe gases along with excess of Ar in different gas samplers which were then estimated using GC. Density Functional Theory (DFT) computations were performed on model Na clusters surrounding Kr/Xe atoms and also on pyrene molecule (a prototype for simulating the spaces within charcoal), trapping Kr and Xe atoms. Experimental observations showed a good correlation with the results obtained from Quantum Theory of Atoms In Molecules (QTAIM), Natural Bond Orbital (NBO), Energy Decomposition (ED), Non-Covalent Interaction (NCI) and Electrostatic Potential (ESP) mapping analyses. Calculated thermochemical parameters affirm the binding of Xe with Na and charcoal to be marginally stronger than that of Kr, consistent with the experimental observations and speculations based on their polarizabilities.

Structural, theoretical and biological activity of mono and binuclear nickel(II) complexes with symmetrical and asymmetrical 4,6-diacetylresorcinol-dithiocarbazate ligands

The present work reports the synthesis and a structural study of two novel dithiocarbazate, the 4,6-diacetylresorcinol-S-benzyldithiocarbazate (H3L1) and the 4,6-diacetylresorcinol-bis(S-benzyldithiocarbazate) (H4L2), and their Ni(II) complexes, [Ni(HL1)(Py)] (1) and [Ni2(L2)(PPh3)2] (2). Single crystal X-ray analyzes reveal mono and binuclear complexes and the metal centers with distorted square planar geometry. The analyses of the Hirshfeld surface and fingerprints plots revealed intermolecular contacts attributed to the H···H and C···H/H···C bonds. The Density Functional Theory (DFT), with the B3LYP functional and 6–311-G(d,p)/LanL2DZ basis sets, was employed to optimize the geometries of synthesized compounds. From the resulting geometries, the highest occupied and lowest unoccupied molecular orbital maps (HOMO-LUMO), orbital energy gap, electron localization function (ELF), electron density, natural bond orbital (NBO) analysis, and complexation of the ligands with Ni(II) were calculated supporting the experimental data. The ESI (+)-MS/MS data indicated the presence in solution of the characteristic fragmentation with the [H3L1]+ and [H4L2]+ molecular ions for the ligands. The pharmacological potential of the dithiocarbazate ligands and their Ni(II) complexes were evaluated in vitro against MDA-MB-231 human breast cancer cells. A remarkable cytotoxic activity was observed, more evident for free ligands than complexes at low concentrations; however, this latter showed a better dose–response pattern, being more attractive in terms of pharmacokinetics and therapeutic window.

Imaging the spatial distribution of π* states in graphene using aberration-corrected and monochromated STEM-EELS: towards orbital mapping

Imaging the spatial distribution of π* states in graphene using aberration-corrected and monochromated STEM-EELS: towards orbital mapping

Bifurcations of hidden orbits in discontinuous maps

One-dimensional maps with discontinuities are known to exhibit bifurcations somewhat different to those of continuous maps. Freed from the constraints of continuity, and hence from the balance of stability that is maintained through fold, flip, and other standard bifurcations, the attractors of discontinuous maps can appear as if from nowhere, and change period or stability almost arbitrarily. But in fact this is misleading, and if one includes states inside the discontinuity in the map, highly unstable ‘hidden orbits’ are created that have iterates on the discontinuity. These populate the bifurcation diagrams of discontinuous maps with just the necessary unstable branches to make them resemble those of continuous maps, namely fold, flip, and other familiar bifurcations. Here we analyse such bifurcations in detail, focussing first on folds and flips, then on bifurcations characterised by creating infinities of orbits, chaotic repellers, and infinite accumulations of sub-bifurcations. We show the role that hidden orbits play, and how they capture the topological structures of continuous maps with steep branches. This suggests both that a more universal dynamical systems theory marrying continuous and discontinuous systems is possible, and shows how discontinuities can be used to approximate steep jumps in continuous systems without losing any of their topological structure.

Crystal structure, Hirshfeld surface analysis, and DFT calculations of methyl (Z)-4-((4-((4-bromobenzyl)selanyl)phenyl)amino)-4-oxobut-2-enoate

The titled organoselenium compound, C18H16BrNO3Se methyl (Z)-4-((4-((4-bromobenzyl)selanyl)phenyl)amino)-4-oxobut-2-enoate (4), was prepared in a three-step synthetic procedure starting from 4,4′-diselanediyldianiline (1). The cytotoxic, antioxidant, and anti-apoptotic properties of the synthesized compounds were evaluated in oligodendrocytes. The crystal structure of 4 shows an orthorhombic system Pbca space group. The two phenyl rings are twisted to each other by 66.70 (15)⁰. The crystal cohesion is established by strong N-H…O hydrogen bonds, N…Br, N….O interactions, and by week aromatic π-π stacking interactions. Hirshfeld surface and the corresponding fingerprint plots of 4 quantify the interactions within the structure. The highest value of the enrichment ratio was calculated, and the Br…N (5.17%) followed by N…O (4%) have a high tendency to form contacts in the crystal. Density functional theory (DFT) optimized structure in the gaseous phase at B3LYP/6-311++G(d,p) level has been compared with the experimentally defined molecular structure. Furthermore, frontier molecular orbital and molecular electrostatic Potential mapping studies have been carried out.

Growth rates of orbits in non-periodic twist maps and a theorem by Neishtadt

We consider non-periodic holomorphic twist maps of the form for α ∈ ]0, 1[ and . Under appropriate assumptions on F 1, F 2 and a primitive of r 1 dθ 1 − r dθ it is shown that , if is a forward complete real orbit of the map.

The Brittle Boulders of Dwarf Planet Ceres

Abstract We mapped all boulders larger than 105 m on the surface of dwarf planet Ceres using images of the Dawn framing camera acquired in the Low Altitude Mapping Orbit. We find that boulders on Ceres are more numerous toward high latitudes and have a maximum lifetime of 150 ± 50 Ma, based on crater counts. These characteristics are distinctly different from those of boulders on asteroid (4) Vesta, an earlier target of Dawn, which implies that Ceres’ boulders are mechanically weaker. Clues to their properties can be found in the composition of Ceres’ complex crust, which is rich in phyllosilicates and salts. As water ice is thought to be present only meters below the surface, we suggest that boulders also harbor ice. Furthermore, the boulder size–frequency distribution is best fit by a Weibull distribution rather than the customary power law, just like for Vesta boulders. This finding is robust in light of possible types of size measurement error.

Fundamental Limits of Map Relative Localization During Spacecraft Descent and Landing

Map relative localization is the process of estimating the location of a lander with respect to the terrain of a celestial body. Visual or topographic maps obtained from orbit are matched with camera or LiDAR data acquired onboard the lander using correlation-based techniques. For given orbital map and descent image resolutions, terrain information content, and measurement noise, there are fundamental limits on the accuracy to which the lander can be localized. Knowledge of these limits is useful in designing and judging the optimality of an estimator, or setting realistic accuracy requirements. This paper derives and validates mathematical expressions that predict these fundamental limits. The derivation, based on application of the Cramér–Rao lower bound, exposes a maximum-likelihood estimator that is confirmed to approach the derived accuracy limits. This work will be useful in the design of future map relative localization systems, as it can be used to inform requirements on desired localization accuracy, and to determine if a near-optimal estimator has been designed.

Feedback control modulation for controlling chaotic maps

Based on existing feedback control methods such as OGY and Pyragas, alternative new schemes are proposed for stabilization of unstable periodic orbits of chaotic and hyperchaotic dynamical systems by suitable modulation of a control parameter. Their performances are improved with respect to: (i) robustness, (ii) rate of convergences, (iii) reduction of waiting time, (iv) reduction of noise sensitivity. These features are analytically investigated, the achievements are rigorously proved and supported by numerical simulations. The proposed methods result successful for stabilizing unstable periodic orbits in some classical discrete maps like 1-D logistic and standard 2-D Hénon, but also in the hyperchaotic generalized n-D Hénon-like maps.

Symmetry type graphs on 4-orbit maps

It is well known that there exist 22 symmetry type graphs associated to [Formula: see text]-orbit maps. For this one, we give the feasible values taken by the degree of the vertices and the number appropriate of edges in the boundary of each face of the map, by introducing the concepts of vertex type graph, face type graph and characteristic system.

The soluble radical and orbits of certain maps on finite groups

For each element u in a finite group G define a map θu:G→G by θu(g)=[g−u,g] and set ΘG(u)={g∈G|θun(g)=g for some n>0}. Then θu induces a permutation of ΘG(u); let βG(u) be the number of orbits apart from {1}. Building on work of J.N. Bray, R.A. Wilson and the second author, we show that the index of the soluble radical of a finite group G is bounded in terms of the values of βG(u) for 2-elements u.

Fluctuations of Time Averages Around Closed Geodesics in Non-Positive Curvature

We consider the geodesic flow for a rank one non-positive curvature closed manifold. We prove an asymptotic version of the Central Limit Theorem for families of measures constructed from regular closed geodesics converging to the Bowen-Margulis-Knieper measure of maximal entropy. The technique expands on ideas of Denker, Senti and Zhang, who proved this type of asymptotic Lindeberg Central Limit Theorem on periodic orbits for expansive maps with the specification property. We extend these techniques from the uniform to the non-uniform setting, and from discrete-time to continuous-time. We consider H\"older observables subject only to the Lindeberg condition and a weak positive variance condition. If we assume a natural strengthened positive variance condition, the Lindeberg condition is always satisfied. Our results extend to dynamical arrays of H\"older observables, and to weighted periodic orbit measures which converge to a unique equilibrium state.

A new method to compute periodic orbits in general symplectic maps

The search of high-order periodic orbits has been typically restricted to problems with symmetries that help to reduce the dimension of the search space. Well-known examples include reversible maps with symmetry lines. The present work proposes a new method to compute high-order periodic orbits in twist maps without the use of symmetries. The method is a combination of the parameterization method in Fourier space and a Newton–Gauss multiple shooting scheme. The parameterization method has been successfully used in the past to compute quasi-periodic invariant circles. However, this is the first time that this method is used in the context of periodic orbits. Numerical examples are presented showing the accuracy and efficiency of the proposed method. The method is also applied to verify the renormalization prediction of the residues’ convergence at criticality (extensively studied in reversible maps) in the relatively unexplored case of maps without symmetries.

In vitro antifungal activities, molecular docking, and DFT studies of 4-amine-3-hydrazino-5-mercapto-1,2,4-triazole derivatives

Six disubstituted Schiff base compounds were synthesized (A1-A6) and characterized using infrared spectroscopy (IR), elemental analyses (EA), 1H NMR, 13C NMR and HRMS spectroscopic techniques. Crystal structure of A1 has been determined by single crystal X-ray diffraction. The antifungal activities against three fungi were assessed, and the results showed that compounds of A1 and A2 have good activity for Wheat gibberellic with EC50 value of 15.89 and 16.99 mg/L, respectively. Compounds of A3, A4 and A6 have good bioactivity against Maize rough bacteria (the value of EC50 is 8.23, 7.56 and 7.92 mg/L, respectively). According to the result of molecular docking, compounds of A1 and A2 have the smallest docking energy (-8.33, −9.00 kcal/mol). Besides, for A1 and A2, the analysis of highest occupied molecular orbital (HOMO), the lowest unoccupied molecular orbital (LUMO) analysis and molecular electrostatic potential map were to further elaborate the reason for the good activity with density functional theory (DFT)-B3LYP/6-31G.

Research on planting arrangement of coconuts in Zhanjiang city under the climate adaptive design

The essential factor that enables people to stay in the active space depends on whether the area can provide comfort to the human body. According to this theory, the climate-adaptive design will effectively improve space vitality. First of all, this paper analysis and summary the climate in Zhanjiang City based on the “golden section method,” obtaining that the comfort of Zhanjiang is dominated by thermal discomfort, and trees can improve the city’s thermal comfort level through shading. Then we established the sun’s orbit map and the trajectory model of coconut shadow changes and used the penetration threshold theory, the following coconut planting arrangement is derived: (1) when the planting quantity is less than or equal to seven or more than 608, there is a reserved space, and the planting arrangement of coconuts is compact; (2) when the number of coconuts is more than seven and less than 608, there will be no reserved space. Currently, the planting arrangement of coconuts is random, and the area of planting space units is less than or equal to 71. 6+x m2 (x is the number of coconuts). These findings will help the designer offer better guidelines for planting and improving the vitality of the space.

The Boulder Population of Asteroid 4 Vesta: Size‐Frequency Distribution and Survival Time

AbstractDawn's framing camera observed boulders on the surface of Vesta when the spacecraftwas in its lowest orbit (Low Altitude Mapping Orbit, LAMO). We identified, measured, and mapped boulders in LAMO images, which have a scale of 20 m per pixel. We estimate that our sample is virtually complete down to a boulder size of 4 pixels (80 m). The largest boulder is a 400 m‐sized block on the Marcia crater floor. Relatively few boulders reside in a large area of relatively low albedo, surmised to be the carbon‐rich ejecta of the Veneneia basin, either because boulders form less easily here or live shorter. By comparing the density of boulders around craters with a known age, we find that the maximum boulder lifetime is about 300 Ma. The boulder size‐frequency distribution (SFD) is generally assumed to follow a power law. We fit power laws to the Vesta SFD by means of the maximum likelihood method, but they do not fit well. Our analysis of power law exponents for boulders on other small Solar System bodies suggests that the derived exponent is primarily a function of boulder size range. The Weibull distribution mimics this behavior and fits the Vesta boulder SFD well. The Weibull distribution is often encountered in rock grinding experiments and may result from the fractal nature of cracks propagating in the rock interior. We propose that, in general, the SFD of particles (including boulders) on the surface of small bodies follows a Weibull distribution rather than a power law.

Bundles with Non-multiplicative Â-Genus and Spaces of Metrics with Lower Curvature Bounds

Abstract We construct smooth bundles with base and fiber products of two spheres whose total spaces have nonvanishing $\hat{A}$-genus. We then use these bundles to locate nontrivial rational homotopy groups of spaces of Riemannian metrics with lower curvature bounds for all ${{\operatorname{Spin}}}$ manifolds of dimension 6 or at least 10, which admit such a metric and are a connected sum of some manifold and $S^n \times S^n$ or $S^n \times S^{n+1}$, respectively. We also construct manifolds $M$ whose spaces of Riemannian metrics of positive scalar curvature have homotopy groups that contain elements of infinite order that lie in the image of the orbit map induced by the push-forward action of the diffeomorphism group of $M$.